This paper proposes a joint uplink massive multiple-input-multiple-output (MIMO) communication and orthogonal frequency-division multiplexing (OFDM) radar sensing architecture. Specifically, uplink communication and short-range radar sensing are considered, where the user equipments (UEs) transmit data to the base-station (BS), which simultaneously receives radar returns from the targets over the same subcarriers. Hence, the signals received at each BS antenna include radar returns and communication signals to be processed for extracting the sensing and communication data. The separation and detection of such signals are achieved by utilizing the channel diversity between the UEs and the targets. To this end, the UEs' signals are first detected and demodulated, and then subtracted from the received signal to acquire the radar returns. Symbol-based radar processing is then employed, as it provides substantial processing gains, and its detection performance is independent of the transmitted radar waveform. Furthermore, self-interference - due to the simultaneous operation of transmit and receive antennas - is taken into account. The communication rate and normalized error of the radar-target channel estimation are mathematically analyzed, and the trade-off between the achievable rate and radar detection performance is demonstrated in terms of the output power of the communication and radar sub-systems and the signal-to-noise ratio.